Planar Groups

Abstract

Abstract. In abstract algebra courses, teachers are often confronted with the task of drawing subgroup lattices. For purposes of instruction, it is usually desirable that these lattices be planar graphs (with no crossings). We present a characterization of abelian groups with this property. We also resolve the following problem in the abelian case: if the subgroup lattice is required to be drawn hierarchically (that is, in monotonic order of index within the group), when is it possible to draw the lattice without crossings?

Finite groups with planar subgroup lattices

“…In [11], Starr and Turner also classify the infinite abelian planar and lattice-planar groups. We know of no examples of infinite nonabelian planar groups.…”

Some Metrical Properties of Lattice Graphs of Finite Groups

“…The lattice graph L(G) of a finite cyclic group G is obtained as follows: Each vertex of L(G) corresponds to an element of H(G), and two vertices corresponding to two elements H 1 , H 2 of H(G) are connected by an edge if and only if H 1 ≤ H 2 and that there is no element K of H(G) such that H 1 K H 2 (see [1,2]), thus ≤ is used when H 1 is proper maximal subgroup of H 2 . The notation ≤ is used as subgroup.…”

Finite Groups Whose Intersection Graphs Are Planar

“…There are interesting graphs constructed from algebraic objects such as the subgroup lattice and the subgroup graph of a group. Planarity of the subgroup lattice and the subgroup graph of a group were studied by Bohanon and Reid in [2] and by Schmidt in [10,11] and by Starr and Turner III in [12], and planarity of the intersection graph of a module over any ring was studied in [13].…”

“…Our methods are elementary. Among the graphs similar to the intersection graphs, we may count the subgroup lattice and the subgroup graph of a group, each of whose planarity was already considered before in [2,10,11,12]. …”